Compositeness for the N* and Δ* resonances from the πn scattering amplitude

Size: px

Start display at page:

Download "Compositeness for the N* and Δ* resonances from the πn scattering amplitude"

Charla Miles
6 years ago
Views:

1 Compositeness for the N* and Δ* resonances from the πn scattering amplitude [1] T. S., Phys. Rev. C95 (2017) [2] T. S., in preparation. Takayasu SEKIHARA (Japan Atomic Energy Agency) 1. Introduction 2. Two-body wave functions from scattering amplitudes 3. The N* compositeness program 4. Summary [3] T. S., T. Hyodo and D. Jido, PTEP D04. [4] T. S., T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) Strangeness and charm in hadrons and dense YITP (May 15-26, 2017)

2 1. Introduction ++ What we have done is ++ For a given interaction (potential) which generates a bound state, we can calculate the wave function of the bound state with the Lippmann-Schwinger Eq. (off-shell scattering amplitude for asymptotic two-body states). --- Not with the Schrödinger Eq. in a usual manner. Furthermore, the wave function from the scattering amplitude is automatically scaled and shows the correct normalization. --- In contrast to the Schrödinger Eq. case, we need not normalize the wave function by hand! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 2

3 1. Introduction ++ What we have done is ++ One can calculate the wave function for a given interaction. --- Seems to be trivial...? Energy dependent interaction. --- Energy dependence of the interaction can be interpreted as a missing-channel contribution. e.g. --> Then the norm of the bound state WF would deviate from unity. Non-relativistic / semi-relativistic kinematics. Stable bound states / unstable resonances. Coupled-channels effect.... These points are clearly explained with the WF from the amplitude. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 3 or

4 2. Wave functions from amplitudes ++ How to calculate the wave function ++ There are several approaches to calculate the wave function. Ex.) A bound state in a NR single-channel problem. Usual approach: Solve the Schrödinger equation. --- Wave function in coordinate / momentum space: --- q > is an eigenstate of free Hamiltonian H0: --> After solving the Schrödinger equation, we have to normalize the wave function by hand. or <-- We require! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 4

5 2. Wave functions from amplitudes ++ How to calculate the wave function ++ There are several approaches to calculate the wave function. Ex.) A bound state in a NR single-channel problem. Our approach: Solve the Lippmann-Schwinger equation at the pole position of the bound state. --- Near the resonance pole position Epole, amplitude is dominated by the pole term in the expansion by the eigenstates of H as --- The residue of the amplitude at the pole position has information on the wave function! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 5

6 2. Wave functions from amplitudes ++ How to calculate the wave function ++ There are several approaches to calculate the wave function. Ex.) The A idea bound of the state renormalization in a NR single-channel for: problem. Our approach: Solve the Lippmann-Schwinger --- equation We (re-)normalize at the pole position the total of wave the bound function state. as cf. --- Near the resonance pole position Epole, amplitude is dominated by the pole term in the expansion by the eigenstates of H as --- The residue of the amplitude at the pole position has information on the wave function! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 6

7 2. Wave functions from amplitudes ++ How to calculate the wave function ++ There are several approaches to calculate the wave function. Ex.) A bound state in a NR single-channel problem. Our approach: Solve the Lippmann-Schwinger equation at the pole position of the bound state. --- The wave function can be extracted from the residue of the amplitude at the pole position: <-- Off-shell Amp.! --> Because the scattering amplitude cannot be freely scaled (Lippmann-Schwinger Eq. is inhomogeneous!), the WF from the residue of the amplitude is automatically scaled as well! If purely molecule --> <-- We obtain! E. Hernandez and A. Mondragon, Phys. Rev. C29 (1984) 722. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 7

8 2. Wave functions from amplitudes ++ Example 1: Stable bound state ++ A Λ hyperon in A ~ 40 nucleus. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψ by hand! Woods-Saxon potential 2. Solve Lippmann-Schwinger equation: --> Extract WF from the residue: --> --- Without normalizing by hand! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 8

9 2. Wave functions from amplitudes ++ Example 1: Stable bound state ++ A Λ hyperon in A ~ 40 nucleus. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψ by hand! Woods-Saxon potential 2. Solve Lippmann-Schwinger equation: --> Extract WF from the residue: --> --- Without normalizing by hand! In 1st way: Points. 2nd way: Lines. Exact coincidence! --- We obtain automatically normalized WF from the Amp.! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 9

10 2. Wave functions from amplitudes ++ Example 1: Stable bound state ++ We define the compositeness X as the norm of the wave function: --- In the following, we calculate X from the scattering amplitude. The compositeness is unity for energy independent interaction. 0s, from Scatt. Amp. X = 1 (v1 = 0) Hernandez and Mondragon (1984). However, if the interaction depends on the energy, the compositeness from the scattering amplitude deviates from unity. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 10

11 2. Wave functions from amplitudes ++ Example 1: Stable bound state ++ We define the compositeness X as the norm of the wave function: --- In the following, we calculate X from the scattering amplitude. The compositeness is unity for energy independent interaction. 0s, from Scatt. Amp. X = 1 (v1 = 0) Lines: X from Amp. Points: X = X V/ E Hernandez and Mondragon (1984). Consistent with the norm with energy-dep. interaction. Formanek, Lombard and Mares (2004); Miyahara and Hyodo (2016). Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 11

12 2. Wave functions from amplitudes ++ Example 1: Stable bound state ++ We define the compositeness X as the norm of the wave function: e.g. --- In the following, we calculate X from the scattering amplitude. The compositeness is unity for energy independent interaction. 0s, from Scatt. Amp. X = 1 (v1 = 0) Hernandez and Mondragon (1984). Deviation of compositeness from unity can be interpreted as a missing-channel part. T. S., Hyodo and Jido, PTEP D04. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 12

13 2. Wave functions from amplitudes ++ Example 2: Unstable resonance state ++ Unstable resonance in KN-πΣ system. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψj by hand! Gaussian potential Coupling strength is controlled by x. 2. Solve Lippmann-Schwinger equation: Aoyama et al. (2006). --> Extract WF from the residue: --> --- Without normalizing by hand! Complex scaling method. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 13

14 2. Wave functions from amplitudes ++ Example 2: Unstable resonance state ++ Unstable resonance in KN-πΣ system. --> Calculate wave functions in 2 ways. 1. Solve Schrödinger equation: --> Normalize ψj by hand! Gaussian potential Coupling strength is controlled by x. 2. Solve Lippmann-Schwinger equation: θ = 20 o --> Extract WF from the residue: --> --- Without normalizing by hand! In 1st way: Points. 2nd way: Lines. Coincidence again! --- Our method is valid even for resonances! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 14

15 2. Wave functions from amplitudes ++ Example 2: Unstable resonance state ++ We define the compositeness X as the norm of the wave function: X Z d 3 q (2 ) 3 h qihq i = Z In the following, we calculate X from the scattering amplitude. <-- The compositeness is unity for energy independent interaction. When we consider the energy dependence of the interaction, the compositeness from the scattering amplitude deviates from unity because of missing channel contribution. --- e.g.: g0 2 V miss = E M 0 0 dq P(q) --- θ Indep.! Hernandez and Mondragon (1984). Lines: X from Amp. Points: X = X V/ E Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 15

2. Wave functions from amplitudes ++ Lessons from schematic models ++ For a given interaction, we can extract the two-body WF from the scattering amplitude at the pole position, both stable and

16 2. Wave functions from amplitudes ++ Lessons from schematic models ++ For a given interaction, we can extract the two-body WF from the scattering amplitude at the pole position, both stable and unstable. Aoyama et al. (2006). T. S., Phys. Rev. C95 (2017) The WF from the scattering amplitude is automatically scaled. The compositeness (= norm of the two-body WF) is unity for a bound state in an energy independent interaction. For an energy dependent interaction, the compositeness deviates from unity, reflecting a missing channel contribution. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 16

17 3. The N* compositeness program ++ What I want to do is ++ For a given interaction, we can calculate two-body wave functions from the scattering amplitude. --- In particular, compositeness (= the norm of the wave function) is automatically normalized! Normalized! Therefore, we can investigate: Compositeness for interesting resonances from amplitudes. Experimental information on the scattering amplitudes available. Construction of detailed interactions possible. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 17

18 3. The N* compositeness program ++ Wave functions for hadrons ++ By using the two-body wave function and compositeness (norm), we can distinguish a certain configuration of hadrons in a model. Hadronic molecules as a bound state of hadrons (cf. deuteron) Ordinary hadrons In the previous studies, we have investigated: Λ(1405). Ξ(1690). N(1535) & N(1650) Evaluated X for these dynamically generated resonances. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 18

3. The N* compositeness program ++ Example: compositeness for Λ(1405) ++

Amplitude taken from: Ikeda, Hyodo and Weise, Phys. Lett. B706, (2011) 63; Nucl.

--- Large KN component for (higher pole) Λ(1405), since XKN is almost unity with

19 3. The N* compositeness program ++ Example: compositeness for Λ(1405) ++ Compositeness X for Λ(1405) in the chiral unitary approach. Amplitude taken from: Ikeda, Hyodo and Weise, Phys. Lett. B706, (2011) 63; Nucl. Phys. A881 (2012) 98.!!! Hyodo and Jido ( 12). --- Large KN component for (higher pole) Λ(1405), since XKN is almost unity with small imaginary parts. T. S., Hyodo and Jido, PTEP 2015, 063D04. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 19

20 3. The N* compositeness program ++ The N* compositeness from πn amplitude ++ Next target: Comprehensive analysis of the N* and Δ* resonances from the precise on-shell πn amplitude! --- The precise on-shell πn scattering amplitude is available. Kamano et al., Phys. Rev. C88 (2014) On-shell scattering amplitude on the real energy E: --- Observable! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 20

3. The N* compositeness program ++ Many N* resonances ++ Many N* and Δ* resonances from the πn scattering amplitude. Suzuki et al., Phys. Rev. Lett. 104 (2010) 042302.

21 3. The N* compositeness program ++ Many N* resonances ++ Many N* and Δ* resonances from the πn scattering amplitude. Suzuki et al., Phys. Rev. Lett. 104 (2010) There are several interesting N* resonances, such as: PDG. We can now investigate their internal structure in terms of the mesonbaryon component. --- N(1440) is a σn bound state? cf. Jülich group. Rönchen et al. (2013);... Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 21

22 3. The N* compositeness program ++ From on-shell to off-shell amplitude ++ By using the on-shell πn amplitude (<-- observable), I construct the off-shell amplitude, where the N* wave functions live. I take into account bare N* states and appropriate diagrams for the meson-baryon interaction. How much the physical N* are dressed? Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 22

23 3. The N* compositeness program ++ Numerical results ++ Numerical results Sorry, but now on going! If you have your own πn amplitudes as solutions of the Lippmann- Schwinger Eq., you can calculate the N* compositeness in the manner presented here. --- Why don t you join me? Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 23

24 We can extract the two-body WF from the residue of the scattering amplitude at the pole position, both stable and unstable states. Scattering amplitude: 4. Summary The WF from the scattering amplitude is automatically scaled. The compositeness (= norm of the two-body WF) is unity for a bound state in an energy independent interaction. For an energy dependent interaction, the compositeness deviates from unity, reflecting a missing channel contribution. From the precise πn amplitude with appropriate models, we can evaluate the compositeness of the N* and Δ* resonances. In particular, how is the structure of the N(1440) resonance? Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 24

25 Thank you very much for your kind attention! Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 25

26 Appendix Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 26

Appendix ++ Compositeness and model (in-)dependence ++ General case: Compositeness are model dependent quantity. Special case: Compositeness for near-threshold poles.

27 Appendix ++ Compositeness and model (in-)dependence ++ General case: Compositeness are model dependent quantity. Special case: Compositeness for near-threshold poles. --- Compositeness can be expressed with threshold parameters such as scattering length and effective range. Deuteron. observables Weinberg ( 65). f0(980) and a0(980). Baru et al. ( 04), Kamiya-Hyodo, Phys. Rev. C93 (2016) Λ(1405). Not observables Kamiya-Hyodo, Phys. Rev. C93 (2016) Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 27

Appendix ++ Compositeness for N(1535) and N(1650) ++ Compositeness X for N(1535) & N(1650) in chiral unitary approach. T. S. T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) 035204.

28 Appendix ++ Compositeness for N(1535) and N(1650) ++ Compositeness X for N(1535) & N(1650) in chiral unitary approach. T. S. T. Arai, J. Yamagata-Sekihara and S. Yasui, Phys. Rev. C93 (2016) For both N* resonances, the missing-channel part Z is dominant. --> N(1535) and N(1650) have large components originating from contributions other than πn, ηn, KΛ, and KΣ. Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 28

29 Appendix ++ Compositeness for Δ(1232) ++ Compositeness X for Δ(1232) in chiral unitary approach.!? The πn compositeness XπN takes large real part! But non-negligible imaginary part as well. --> Large πn component in the Δ(1232) resonance!? Strangeness and charm in hadrons and dense YITP (May 15-26, 2017) 29

Compositeness of Dynamically Generated Resonances

Compositeness of Dynamically Generated Resonances Takayasu SEKIHARA (Japan Atomic Energy Agency) [1] T. S., Phys. Rev. C95 (2017) 025206. [2] T. S., T. Hyodo and D. Jido, PTEP 2015 063D04. [3] T. S., T.